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Relative Proj
In algebraic geometry, Proj is a construction analogous to the spectrum-of-a-ring construction of affine schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not functorial, is a fundamental tool in scheme theory. In this article, all rings will be assumed to be commutative and with identity. Proj of a graded ring Proj as a set Let S be a commutative graded ring, whereS = \bigoplus_ S_iis the direct sum decomposition associated with the gradation. The irrelevant ideal of S is the ideal of elements of positive degreeS_+ = \bigoplus_ S_i .We say an ideal is homogeneous if it is generated by homogeneous elements. Then, as a set,\operatorname S = \. For brevity we will sometimes write X for \operatorname S. Proj as a topological space We may define a topology, called the Zariski topology, on \operatorname S by defining the closed sets to be those of the form :V(a) = \, where a is a homogeneous idea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects. The fundamental objects of study in algebraic geometry are algebraic variety, algebraic varieties, which are geometric manifestations of solution set, solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are line (geometry), lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscate of Bernoulli, lemniscates and Cassini ovals. These are plane algebraic curves. A point of the plane lies on an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of points of special interest like singular point of a curve, singular p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
